



'''simple docstring'''



from __future__ import annotations
def        UpperCAmelCase (    A	:					list[int]       ,		A	:					list[int]       ,		A	:					list[int]       ,		A	:					list[list[str]]       ,		A	:					int       ,		):
							SCREAMING_SNAKE_CASE       :   Optional[int]         =      len(A					)

							# If row is equal to the size of the board it means there are a queen in each row in
							# the current board (possible_board)
							if row == n:
														# We convert the variable possible_board that looks like this: [1, 3, 0, 2] to
														# this: ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
														boards.append(['''. ''' * i + '''Q ''' + '''. ''' * (n - 1 - i) for i in possible_board]					)
														return

							# We iterate each column in the row to find all possible results in each row
							for col in range(A					):
														# We apply that we learned previously. First we check that in the current board
														# (possible_board) there are not other same value because if there is it means
														# that there are a collision in vertical. Then we apply the two formulas we
														# learned before:
														#
														# 45º: y - x = b or 45: row - col = b
														# 135º: y + x = b or row + col = b.
														#
														# And we verify if the results of this two formulas not exist in their variables
														# respectively.  (diagonal_right_collisions, diagonal_left_collisions)
														#
														# If any or these are True it means there is a collision so we continue to the
														# next value in the for loop.
														if (
														    col in possible_board
														    or row - col in diagonal_right_collisions
														    or row + col in diagonal_left_collisions
														):
																					continue

														# If it is False we call dfs function again and we update the inputs
														depth_first_search(
														    [*possible_board, col]       ,		[*diagonal_right_collisions, row - col]       ,		[*diagonal_left_collisions, row + col]       ,		A       ,		A       ,		)
def        UpperCAmelCase (    A	:					int					):
							SCREAMING_SNAKE_CASE       :   list[list[str]]         =      []
							depth_first_search([]       ,		[]       ,		[]       ,		A       ,		A					)

							# Print all the boards
							for board in boards:
														for column in board:
																					print(A					)
														print(''''''					)

							print(len(A					)       ,		'''solutions were found.'''					)


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							n_queens_solution(4)









'''simple docstring'''



from queue import PriorityQueue
from typing import Any

import numpy as np
def        UpperCAmelCase (    A	:					dict       ,		A	:					str       ,		A	:					set       ,		A	:					set       ,		A	:					dict       ,		A	:					dict       ,		A	:					PriorityQueue       ,		A	:					dict       ,		A	:					float | int       ,		):
							for nxt, d in graph[v]:
														if nxt in visited_forward:
																					continue
														SCREAMING_SNAKE_CASE       :   str         =      cst_fwd.get(A       ,		np.inf					)
														SCREAMING_SNAKE_CASE       :   Optional[int]         =      cst_fwd[v] + d
														if new_cost_f < old_cost_f:
																					queue.put((new_cost_f, nxt)					)
																					SCREAMING_SNAKE_CASE       :   Tuple         =      new_cost_f
																					SCREAMING_SNAKE_CASE       :   List[str]         =      v
														if nxt in visited_backward:
																					if cst_fwd[v] + d + cst_bwd[nxt] < shortest_distance:
																												SCREAMING_SNAKE_CASE       :   Union[str, Any]         =      cst_fwd[v] + d + cst_bwd[nxt]
							return shortest_distance
def        UpperCAmelCase (    A	:					str       ,		A	:					str       ,		A	:					dict       ,		A	:					dict					):
							SCREAMING_SNAKE_CASE       :   Dict         =      -1

							SCREAMING_SNAKE_CASE       :   Optional[int]         =      set()
							SCREAMING_SNAKE_CASE       :   Optional[Any]         =      set()
							SCREAMING_SNAKE_CASE       :   Any         =      {source: 0}
							SCREAMING_SNAKE_CASE       :   Tuple         =      {destination: 0}
							SCREAMING_SNAKE_CASE       :   Union[str, Any]         =      {source: None}
							SCREAMING_SNAKE_CASE       :   Optional[int]         =      {destination: None}
							SCREAMING_SNAKE_CASE       :   PriorityQueue[Any]         =      PriorityQueue()
							SCREAMING_SNAKE_CASE       :   PriorityQueue[Any]         =      PriorityQueue()

							SCREAMING_SNAKE_CASE       :   List[str]         =      np.inf

							queue_forward.put((0, source)					)
							queue_backward.put((0, destination)					)

							if source == destination:
														return 0

							while not queue_forward.empty() and not queue_backward.empty():
														SCREAMING_SNAKE_CASE       ,SCREAMING_SNAKE_CASE       :   str         =      queue_forward.get()
														visited_forward.add(A					)

														SCREAMING_SNAKE_CASE       ,SCREAMING_SNAKE_CASE       :   int         =      queue_backward.get()
														visited_backward.add(A					)

														SCREAMING_SNAKE_CASE       :   Tuple         =      pass_and_relaxation(
														    A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		)

														SCREAMING_SNAKE_CASE       :   Optional[int]         =      pass_and_relaxation(
														    A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		A       ,		)

														if cst_fwd[v_fwd] + cst_bwd[v_bwd] >= shortest_distance:
																					break

							if shortest_distance != np.inf:
														SCREAMING_SNAKE_CASE       :   int         =      shortest_distance
							return shortest_path_distance


lowerCAmelCase_  :							int             =    {
    'B': [['C', 1]],
    'C': [['D', 1]],
    'D': [['F', 1]],
    'E': [['B', 1], ['G', 2]],
    'F': [],
    'G': [['F', 1]],
}
lowerCAmelCase_  :							str             =    {
    'B': [['E', 1]],
    'C': [['B', 1]],
    'D': [['C', 1]],
    'F': [['D', 1], ['G', 1]],
    'E': [[None, np.inf]],
    'G': [['E', 2]],
}

if __name__ == "__main__":
							import doctest

							doctest.testmod()




